A random sample of n = 9 scores is selected from a normal distribution with μ = 80 and σ = 12.What is the probability that the sample mean will be between 76 and 84?
A)0.9974
B)0.6826
C)0.3830
D)0.2586

Question

Quizplus · Accepted Answer

We can use the central limit theorem to approximate the distribution of sample means as normal with mean μ and standard deviation σ/sqrt(n). 
Thus, for this problem, we have:
mean = μ = 80
standard deviation = σ/sqrt(n) = 12/sqrt(9) = 4
Now, we need to find the probability that the sample mean, x̄, is between 76 and 84. 
To do this, we standardize the values:
z1 = (76-80)/4 = -1
z2 = (84-80)/4 = 1
Then, we use a normal distribution table or calculator to find the probability that a standard normal variable is between -1 and 1 (i.e., the area under the curve between z1 and z2). This probability is approximately 0.6826. Therefore, the answer is B.

A Random Sample of N = 9 Scores Is Selected